An additional projection step to He and Liao's method for solving variational inequalities
Journal of Computational and Applied Mathematics
Nonmonotone projected gradient methods based on barrier and Euclidean distances
Computational Optimization and Applications
Modified extragradient methods for solving variational inequalities
Computers & Mathematics with Applications
The interior proximal extragradient method for solving equilibrium problems
Journal of Global Optimization
Inexact Proximal Point Methods for Variational Inequality Problems
SIAM Journal on Optimization
Weighted variational inequalities in non-pivot Hilbert spaces with applications
Computational Optimization and Applications
Korpelevich's method for variational inequality problems in Banach spaces
Journal of Global Optimization
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We propose new interior projection type methods for solving monotone variational inequalities. The methods can be viewed as a natural extension of the extragradient and hyperplane projection algorithms, and are based on using non Euclidean projection-like maps. We prove global convergence results and establish rate of convergence estimates. The projection-like maps are given by analytical formulas for standard constraints such as box, simplex, and conic type constraints, and generate interior trajectories. We then demonstrate that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem. As a consequence, the algorithms are easy to implement, with low computational cost, and naturally lead to decomposition schemes for problems with a separable structure. This is illustrated through examples for convex programming, convex-concave saddle point problems and semidefinite programming.